Pathwise asymptotic behavior of random determinants in the uniform Gram and Wishart ensembles

الأصل · EN

This paper concentrates on asymptotic properties of determinants of some random symmetric matrices. If Bₙ,ᵣ is a n x r rectangular matrix and Bₙ,ᵣ' its transpose, we study det (Bₙ,ᵣ'Bₙ,ᵣ) when n,r tends to infinity with r/n → c∈ (0,1). The r column vectors of Bₙ,ᵣ are chosen independently, with common distribution νₙ. The Wishart ensemble corresponds to νₙ = N(0, Iₙ), the standard normal distribution. We call uniform Gram ensemble the ensemble corresponding to νₙ = σₙ, the uniform distribution on the unit sphere `Sₙ₋₁. In the Wishart ensemble, a well known Bartlett's theorem decomposes the above determinant into a product of chi-square variables. The same holds in the uniform Gram ensemble. This allows us to study the process {1/n (Bn, nt'Bn, nt), t ∈ [0,1]} and its asymptotic behavior as n→ ∞: a.s. convergence, fluctuations, large deviations. We connect the results for marginals (fixed t) with those obtained by the spectral method.